Post on 07-Nov-2021
Grade 2
Curriculum Guide
SY 2011-12 through SY 2017-18
Mathematics
Mathematics Curriculum Guide Introduction Prince William County Schools
The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying Essential Understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn. The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, and Teacher Notes and Elaborations. Resources and Sample Instructional Strategies and Activities are included in the Unit Guides. The purpose of each section is explained below. Curriculum Information: This section includes the objective and SOL Reporting Category. Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be either an exhaustive list or a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective. Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills. Essential Questions and Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning.
FOCUS K-3_______________STRAND: NUMBER AND NUMBER SENSE___________GRADE LEVEL 2
Students in grades K–3 have a natural curiosity about their world, which leads them to develop a sense of number. Young children are
motivated to count everything around them and begin to develop an understanding of the size of numbers (magnitude), multiple ways
of thinking about and representing numbers, strategies and words to compare numbers, and an understanding of the effects of simple
operations on numbers. Building on their own intuitive mathematical knowledge, they also display a natural need to organize things
by sorting, comparing, ordering, and labeling objects in a variety of collections.
Consequently, the focus of instruction in the number and number sense strand is to promote an understanding of counting,
classification, whole numbers, place value, fractions, number relationships (―more than,‖ ―less than,‖ and ―equal to‖), and the effects
of single-step and multistep computations. These learning experiences should allow students to engage actively in a variety of
problem solving situations and to model numbers (compose and decompose), using a variety of manipulatives. Additionally, students
at this level should have opportunities to observe, to develop an understanding of the relationship they see between numbers, and to
develop the skills to communicate these relationships in precise, unambiguous terms.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Number and Number Sense
Virginia SOL 2.1
The student will
a. read, write, and identify the place
value of each digit in a three-digit numeral, using numeration models;
b. round two-digit numbers to the
nearest ten; and
c. compare two whole numbers
between 0 and 999, using symbols
(>, <, or =) and words (is greater
than, is less than, or is equal to).
Foundational Objective
1.1 The student will
b. group a collection of up to 100
objects into tens and ones and write
the corresponding numeral to
develop an understanding of place
value.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Demonstrate the understanding of the
ten-to-one relationships among ones,
tens, and hundreds, using manipulatives
(e.g., beans and cups, Base-10 blocks,
bundles of ten sticks).
Determine the place value of each digit in a three-digit numeral presented as a
pictorial representation (e.g., a picture
of Base-10 blocks) or as a physical rep-
resentation (e.g., actual Base-10 blocks).
Write numerals, using a Base-10 model
or picture.
Read three-digit numbers when shown
a numeral, a Base-10 model of the
number, or a pictorial representation of
the number.
Identify the place value (ones, tens, hundreds) of each digit in a three-digit
numeral.
Determine the value of each digit in a
three-digit numeral (e.g., in 352, the 5
represents 5 tens and its value is 50).
Round two-digit numbers to the nearest
ten.
Compare two numbers between 0 and
999 represented pictorially or with
concrete objects (e.g., Base-10 blocks),
using the words is greater than, is less than, or is equal to.
Key Vocabulary compare ones
hundreds place value
is equal to (=) round
is greater than (>) tens
is less than (<)
Essential Questions
How do patterns in our place value number system help us read, write, and compare
whole numbers?
How can we use models to demonstrate the value of each digit in a two- or three-digit
number?
What does it mean to round numbers to the nearest 10? Why is rounding numbers
useful?
What words and symbols are used to compare whole numbers?
Essential Understandings
All students should
Understand the ten-to-one relationship of ones, tens, and hundreds (10 ones equals 1 ten,
10 tens equals 1 hundred).
Understand that numbers are written to show how many hundreds, tens, and ones are in
the number.
Understand that rounding gives a close, easy-to-use number to use when an exact
number is not needed for the situation at hand.
Understand that knowledge of place value is essential when comparing numbers.
Understand the relative magnitude of numbers by comparing numbers.
Teacher Notes and Elaborations The number system is based on a simple pattern of tens where each place has ten times the
value of the place to its right.
Opportunities to experience the relationships among hundreds, tens, and ones through
hands-on experiences with manipulatives are essential to developing the ten-to-one place
value concept of our number system and understanding the value of each digit in a three-
digit number. Ten-to-one trading activities with manipulatives on place value mats provide
excellent experiences to develop the understanding of the places in the Base-10 system.
Place value charts should then be used with number cards or number tiles. Students should
build the numbers being compared on their place value charts. This should be related to the
Base-10 blocks. Students should begin to understand that they need to compare the larger places first.
Models that clearly illustrate the relationships among hundreds, tens, and ones are
physically proportional (e.g., the tens piece is ten times larger than the ones piece).
Students need to understand 10 and 100 as special units of numbers (e.g., 10 is 10 ones, but
it is also 1 ten).
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Number and Number Sense
Virginia SOL 2.1
The student will
a. read, write, and identify the place
value of each digit in a three-digit numeral, using numeration models;
b. round two-digit numbers to the
nearest ten; and
c. compare two whole numbers
between 0 and 999, using symbols
(>, <, or =) and words (is greater
than, is less than, or is equal to).
Foundational Objective
1.1 The student will
b. group a collection of up to 100
objects into tens and ones and write
the corresponding numeral to
develop an understanding of place
value.
Teacher Notes and Elaborations (continued)
Flexibility in thinking about numbers is critical. Students should understand that 123 may be thought of as 123 ones; or 1 hundred, 2 tens,
and 3 ones; or 12 tens and 3 ones.
Rounding is finding the nearest easy-to-use number (e.g., the nearest 10) for the situation at hand. Emphasis should be on understanding
the rounding concept, not on memorization of a procedure. Students should develop the procedure for rounding instead of memorizing the
procedure without understanding.
Number lines are useful tools for developing the concept of rounding to the nearest ten. Rounding to the nearest ten using a number line is done as follows:
– Locate the number on the number line.
– Identify the two tens between which the number lies.
– Determine the closest ten.
– If the digit in the ones place is 5 (halfway between the two tens), round the number to the higher ten.
Once the concept for rounding numbers using a number line is developed, the procedure for
rounding numbers to the nearest ten is as follows:
– Look one place to the right of the digit in the place you wish to round.
– If the digit is less than 5, leave the digit in the rounding place as it is, and change the digits to the right of the rounding place to
zero.
– If the digit is 5 or greater, add 1 to the digit in the rounding place and change the digits to the right of the rounding place to zero.
A procedure for comparing two numbers by examining place value may include the following:
– Line up the numbers by place value lining up the ones.
– Beginning at the left, find the first place where the digits are different.
– Compare the digits in this place value to determine which number is greater (or which is less).
– Use the appropriate symbol > or < or words is greater than or is less than to compare the numbers in the order in which they are
presented.
– If both numbers are the same, use the symbol = or the words is equal to.
Mathematical symbols (>, <) used to compare two unequal numbers are called inequality symbols.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Number and Number Sense
Virginia SOL 2.2
The student will
a. identify the ordinal positions first
through twentieth, using an ordered set of objects; and
b. write the ordinal numbers.
Foundational Objective
1.1
The student will
a. count from 1 to 100 and write the
corresponding numeral.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Count an ordered set of objects using
ordinal number words first through
twentieth.
Identify ordinal positions first through
twentieth, using an ordered set of
objects.
Identify ordinal positions first through
twentieth, using an ordered set of
objects presented in lines, or rows from
– left to right;
– right to left;
– top to bottom; and
– bottom to top.
Write1st, 2nd, 3rd, through 20th in
numerals.
Key Vocabulary
eighth eighteenth
eleventh fifteenth
fifth first
fourteenth fourth
nineteenth ninth
ordinal second
seventeenth seventh
sixteenth sixth
tenth third
thirteenth twelfth twentieth
Essential Questions
How do ordinal numbers help us identify items?
How are ordinal numbers named and written? How are they different from counting
numbers?
How can we use objects to model the ordinal positions from the first to the twentieth
positions?
Essential Understandings
All students should
Use ordinal numbers to describe the position of an object in a sequence or set.
Teacher Notes and Elaborations
Understanding the cardinal and ordinal meaning of numbers is necessary to quantify,
measure, and identify the order of objects.
An ordinal number is a number that names the place or position of an object in sequence or
set (e.g., first, third). Ordered position, ordinal position, and ordinality are terms that refer
to the place or position of an object in a sequence or a set.
The ordinal position is determined by where one starts in an ordered set of objects or
sequence of objects (e.g., left, right, top, bottom).
The ordinal meaning of numbers is developed by identifying and verbalizing the place or
position of objects in a set or sequence (e.g., the student’s position in line when students are
lined up alphabetically by their first name).
Ordinal position can also be emphasized through sequencing events (e.g., months in a year,
sequencing in a story).
Cardinality can be compared with ordinality when comparing the results of counting. There
is obvious similarity between the ordinal number words third through twentieth and the
cardinal number words three through twenty.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Number and Number Sense
Virginia SOL 2.3
The student will
a. identify the parts of a set and/or
region that represent fractions for halves, thirds, fourths, sixths,
eighths, and tenths;
b. write the fractions; and
c. compare the unit fractions for
halves, thirds, fourths, sixths,
eighths, and tenths.
Foundational Objectives
1.3
The student will identify the parts of a set and/or region that represent
fractions for halves, thirds, and fourths
and write the fractions.
K.5
The student will identify the parts of a
set and/or region that represent
fractions for halves and fourths.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Recognize fractions as representing
equal-size parts of a whole.
Identify the fractional parts of a whole
or a set for22 ,
32 ,
43 ,
62 ,
87 ,
107 , etc.
Identify the fraction names (halves,
thirds, fourths, sixths, eighths, and
tenths) for the fraction notation
22 ,
32 ,
43 ,
62 ,
87 ,
107 , etc.
Represent fractional parts of a whole
for halves, thirds, fourths, sixths,
eighths, tenths using
– region/area models (i.e. pie pieces,
pattern blocks, geo boards);
– sets (e.g., chips, counters, cubes); and
– measurement models (e.g., fraction
strips, rods, connecting cubes).
Compare unit fractions (21 ,
31 ,
41 ,
61 ,
81 ,
and 101 using the words is greater than,
is less than, or is equal to and the
symbols (>, <, =).
Key Vocabulary
eighths
fraction
fourths
halves
tenths
thirds
part
sixths
unit fraction
whole
Essential Questions
How can we model fractional parts of a region or area? How are the parts identified?
How can we model fractional parts of a set? How are the parts identified?
What do we need to think about when we compare fractions or put them in size order?
Why are the words part, whole, and equal important when working with fractions?
Essential Understandings
All students should
Understand that fractional parts are equal shares of a whole or a whole set.
Understand that the fraction name (half, fourth) tells the number of equal parts in the whole.
Understand that when working with unit fractions, the larger the denominator, the
smaller the part and therefore the smaller the fraction.
Teacher Notes and Elaborations
The whole should be defined.
A fraction is a way of representing part of a whole (as in a region/area model) or part of a
group (as in a set model). Fractional parts are equal shares or equal-sized portions of a
whole or unit.
In each fraction model, the parts must be equal in size (i.e., the size of each pie piece must
have the same area; the size of each chip in a set must be equal), or the parts must be equal
in number (i.e., if there are four different toys and John has one of them, he has 41 of the set
of toys even though they are not the same size.)
Students should have experiences dividing a whole into additional parts. As the whole is
divided into more parts, students understand that each part becomes smaller.
The denominator tells how many equal parts are in the whole or set. The numerator tells
how many of those parts are being described. These words (denominator and numerator) are
not required at this grade, but the concept of parts and wholes is required for understanding
of a fraction.
Students should have opportunities to make connections among fraction representations by
connecting the concrete or pictorial representations with spoken or symbolic
representations.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Number and Number Sense
Virginia SOL 2.3
The student will
a. identify the parts of a set and/or
region that represent fractions for halves, thirds, fourths, sixths,
eighths, and tenths;
b. write the fractions; and
c. compare the unit fractions for
halves, thirds, fourths, sixths,
eighths, and tenths.
Foundational Objectives
1.3
The student will identify the parts of a set and/or region that represent
fractions for halves, thirds, and fourths
and write the fractions.
K.5
The student will identify the parts of a
set and/or region that represent
fractions for halves and fourths.
Teacher Notes and Elaborations (continued)
Informal, integrated experiences with fractions at this level will help students develop a foundation for deeper learning at later grades.
Understanding the language of fractions (e.g., such as thirds means ―three equal parts of a whole‖ or 31 represents one of three equal-size
parts when a pizza is shared among three students) will further this development.
A unit fraction is one in which the numerator is one. Using models when comparing unit fractions will assist in developing the concept that
the larger the denominator, the smaller the piece; therefore, 41 <
31 .
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Number and Number Sense
Virginia SOL 2.4
The student will
a. count forward by twos, fives, and
tens to 100, starting at various multiples of 2, 5, or 10;
b. count backward by tens from 100;
and
c. recognize even and odd numbers.
Foundational Objectives
1.2
The student will count forward by ones,
twos, fives, and tens to 100 and
backward by ones from 30.
K.4
The student will
a. count forward to 10 and backward
from 10;
b. identify one more than a number
and one less than a number; and
c. count by fives and tens to 100.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Determine patterns created by counting
by twos, fives, and tens on a hundred
chart.
Skip count by twos, fives, and tens to
100 using manipulatives, a hundred
chart, mental mathematics, a calculator, and/or paper and pencil.
Skip count by twos, fives, and tens to
100.
Count backward by tens from 100.
Use objects to determine whether a
number is odd or even.
Key Vocabulary
backward
even fives
forward
odd
skip counting
tens
twos
Essential Questions
How can we use tools (objects, number lines, hundred charts, and calculators) to help us
find patterns in numbers?
How can patterns in our number system help us skip count by 2s, 5s, and 10s, (forward
or backward) no matter what number we start with?
Where are skip-counting patterns found in our everyday lives?
What patterns are formed by even and odd numbers? How can we use pairing to
demonstrate that a number is odd or even?
Essential Understandings
All students should
Understand that a collection of objects can be grouped and skip counting can be used to
count the collection.
Describe the patterns in skip counting and use those patterns to predict the next number
in the counting sequence.
Understand that the starting point for skip counting by 2 does not always begin at 2.
Understand that the starting point for skip counting by 5 does not always begin at 5.
Understand that the starting point for skip counting by 10 does not always begin at 10.
Understand that every counting number is either even or odd.
Teacher Notes and Elaborations The patterns developed as a result of grouping and/or skip-counting are precursors for
recognizing numeric patterns, functional relationships, and concepts underlying money,
telling time, and multiplication and division. Powerful models for developing these
concepts include counters, hundred chart, and calculators.
Skip counting by twos supports the development of the concept of even numbers. Skip
counting by twos starting with one produces the odd numbers. Skip counting by fives lays
the foundation for reading a clock effectively and telling time to the nearest five minutes,
counting money, and developing the multiplication facts for five. Skip counting by tens is
a precursor for use of place value, addition, counting money, and multiplying by multiples
of ten.
Calculators can be used to display the numeric patterns resulting from skip counting. Use
the constant feature of the four-function calculator to display the numbers in the sequence
when skip counting by that constant.
Odd and even numbers can be explored in different ways (e.g., dividing collections of
objects into two equal groups or pairing objects).
FOCUS K-3_____________STRAND: COMPUTATION AND ESTIMATION_________GRADE LEVEL 2
A variety of contexts are necessary for children to develop an understanding of the meanings of the operations such as addition and
subtraction. These contexts often arise from practical experiences in which they are simply joining sets, taking away or separating
from a set, or comparing sets. These contexts might include conversations, such as ―How many books do we have altogether?‖ or
―How many cookies are left if I eat two?‖ or ―I have three more candies than you do.‖ Although young children first compute using
objects and manipulatives, they gradually shift to performing computations mentally or using paper and pencil to record their thinking.
Therefore, computation and estimation instruction in the early grades revolves around modeling, discussing, and recording a variety of
problem situations. This approach helps students transition from the concrete to the representation to the symbolic in order to develop
meaning for the operations and how they relate to each other.
In grades K–3, computation and estimation instruction focuses on
relating the mathematical language and symbolism of operations to problem situations;
understanding different meanings of addition and subtraction of whole numbers and the relation between the two operations;
developing proficiency with basic addition, subtraction, multiplication, division and related facts;
gaining facility in manipulating whole numbers to add and subtract and in understanding the effects of the operations on whole
numbers;
developing and using strategies and algorithms to solve problems and choosing an appropriate method for the situation;
choosing, from mental computation, estimation, paper and pencil, and calculators, an appropriate way to compute;
recognizing whether numerical solutions are reasonable;
experiencing situations that lead to multiplication and division, such as equal groupings of objects and sharing equally; and
performing initial operations with fractions.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Computation and Estimation
Virginia SOL 2.5
The student will recall addition facts
with sums to 20 or less and the
corresponding subtraction facts.
Foundational Objectives
1.5
The student will recall basic addition
facts with sums to 18 or less and the
corresponding subtraction facts.
K.6
The student will model adding and
subtracting whole numbers, using up to10 concrete items.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Recall and write the basic addition facts
for sums to 20 or less and the
corresponding subtraction facts, when
addition and subtraction problems are
presented in either horizontal or vertical
written format.
Key Vocabulary
adding
addition
combine
difference
equal
minus
plus related facts
separate
subtracting
subtraction
sum
Essential Questions
How is addition like combining?
How is subtracting like separating?
What are different ways to write an addition or subtraction fact?
How can models (snap cubes, ten frames, Base-10 materials) help us ―see‖ a fact in our
minds?
What are some strategies for learning basic addition and subtraction facts?
Essential Understandings
All students should
Understand that addition involves combining and subtraction involves separating.
Develop fluency in recalling facts for addition and subtraction.
Teacher Notes and Elaborations
Associate the terms addition, adding, and sum with the concept of joining or combining.
Associate the terms subtraction, subtracting, minus, and difference with the process of
―taking away‖ or separating (i.e., removing a set of objects from the given set of objects;
finding the difference between two numbers; or comparing two numbers).
Provide practice in the use and selection of strategies. Encourage students to develop efficient strategies. Examples of strategies for developing the basic addition and subtraction
facts include:
- counting on;
- counting back;
- ―one-more-than‖, ― two-more-than‖ facts;
- ―one-less-than‖, ―two-less-than‖ facts;
- ―doubles‖ to recall addition facts (e.g., 2 + 2 =___ ; 3 + 3 =___ );
- ―near doubles‖ [e.g., 3 + 4 = (3 + 3) + 1 = ____];
- ―make ten‖ facts (e.g., at least one addend of 8 or 9);
- ―think addition for subtraction,‖ (e.g., for 9 – 5 = ____; think ―5 and what number
makes 9?‖);
- use of the commutative property without naming the property (e.g., 4 +3 is the same as 3 + 4);
- use of related facts (e.g., 4 + 3 = 7, 3 + 4 = 7, 7 – 4 = 3, and 7 – 3 = 4); and
- use of the additive identity property (e.g., 4+0=4), without naming the property but
saying ―When you add zero to a number, you always get the original number.‖
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Computation and Estimation
Virginia SOL 2.5
The student will recall addition facts
with sums to 20 or less and the
corresponding subtraction facts.
Foundational Objectives
1.5
The student will recall basic addition
facts with sums to 18 or less and the corresponding subtraction facts.
K.6
The student will model adding and
subtracting whole numbers, using up
to10 concrete items.
Teacher Notes and Elaborations (continued)
Manipulatives should be used initially to develop an understanding of the addition and subtraction facts and to engage students in
meaningful memorization. Rote recall of the facts is often achieved through constant practice and may come in a variety of formats,
including presentation through counting on, related facts, flash cards, practice sheets, and/or games.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Computation and Estimation
Virginia SOL 2.6
The student, given two whole numbers
whose sum is 99 or less, will
a. estimate the sum; and b. find the sum, using various methods
of calculation.
Foundational Objective
1.4
The student, given a familiar problem
situation involving magnitude, will
a. select a reasonable order of
magnitude from three given
quantities: a one-digit numeral, a two-digit numeral, and a three-digit
numeral (e.g., 5, 50, 500); and
b. explain the reasonableness of the
choice.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Regroup 10 ones for 1 ten, using Base-
10 models, when finding the sum of
two whole numbers whose sum is
99 or less.
Estimate the sum of two whole numbers
whose sum is 99 or less and recognize whether the estimation is reasonable.
Find the sum of two whole numbers
whose sum is 99 or less, using Base-10
blocks and bundles of tens.
Solve problems presented vertically or
horizontally that require finding the
sum of two whole numbers whose sum
is 99 or less, using paper and pencil.
Solve problems involving addition of
two whole numbers whose sum is 99 or
less by using mental computation strategies.
Key Vocabulary
estimate
join
Essential Questions
How can we find sums using Base-10 models?
When is an estimate more useful than an exact sum? What are some strategies to
estimate sums?
Why is it important to estimate first, before calculating a sum?
What are different strategies to compute sums? How do we decide which to use?
What strategies help us compute sums mentally?
Essential Understandings
All students should
Understand that estimation skills are valuable, timesaving tools particularly in practical
situations when exact answers are not required or needed.
Understand that estimation skills are also valuable in determining the reasonableness of
the sum when solving for the exact answer is needed.
Understand that addition is used to join groups in practical situations when exact
answers are needed.
Develop flexible methods of adding whole numbers by combining numbers in a variety
of ways to find the sum, most depending on place values.
Teacher Notes and Elaborations Addition means to combine or join quantities.
The terms used in addition are
23 addend
+ 46 + addend
69 sum
Estimation is a number sense skill used instead of finding an exact answer. When an actual
computation is not necessary, an estimate will suffice.
Rounding addends is one strategy used to estimate.
Estimation is also used before solving a problem to check the reasonableness of the sum
when an exact answer is required.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Computation and Estimation
Virginia SOL 2.6
The student, given two whole numbers
whose sum is 99 or less, will
a. estimate the sum; and b. find the sum, using various methods
of calculation.
Foundational Objective
1.4
The student, given a familiar problem
situation involving magnitude, will
c. select a reasonable order of
magnitude from three given
quantities: a one-digit numeral, a two-digit numeral, and a three-digit
numeral (e.g., 5, 50, 500); and
d. explain the reasonableness of the
choice.
Teacher Notes and Elaborations (continued)
By estimating the result of an addition problem, a place value orientation for the answer is established.
The traditional algorithm for addition of two-digit numbers is contrary to the natural inclination to begin with the left-hand number. An
alternative recording method shown below may make much more sense to a student than the traditional method of recording. In this
recording scheme it does not matter whether the student begins with the digits on the left or the right. This thinking is also conducive to
mental addition of numbers since it involves the use of place value.
Alternative recording method:
24
+39
50 (30 + 20)
13 (9 + 4)
63
Strategies for mentally adding two-digit numbers include student-invented strategies, making-ten, partial sums, and counting on, among
others.
partial sums: 56 + 41 = ?
50 + 40 = 90
6 + 1 = 7 90 + 7 = 97
counting on: 36 + 62 = ?
36 + 60 = 96
96 + 2 = 98
Strategies for adding two-digit numbers can include, but are not limited to, using a hundred chart, number line, and invented strategies.
Regrouping is used in addition when a sum in a particular place is 10 or greater.
Building an understanding of the algorithm by first using concrete materials and then a ―do-and-write‖ approach connects it to the written form of the algorithm.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand
Computation and Estimation
Virginia SOL 2.7
The student, given two whole numbers,
each of which is 99 or less, will
a. estimate the difference; and b. find the difference, using various
methods of calculation.
Foundational Objectives
1.4
The student, given a familiar problem
situation involving magnitude, will
e. select a reasonable order of
magnitude from three given
quantities: a one-digit numeral, a two-digit numeral, and a three-digit
numeral (e.g., 5, 50, 500); and
f. explain the reasonableness of the
choice.
K.6
The student will model adding and
subtracting whole numbers, using up
to10 concrete items.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Regroup 1 ten for 10 ones, using Base-
10 models, such as Base-10 blocks and
bundles of tens.
Estimate the difference of two whole
numbers each 99 or less and recognize
whether the estimation is reasonable.
Find the difference between two whole
numbers each 99 or less, using Base-10
models, such as Base-10 blocks and
bundles of tens.
Solve problems presented vertically or
horizontally that require finding the
difference of two whole numbers each
99 or less, using paper and pencil.
Solve problems involving subtraction of
two whole numbers, each 99 or less,
using mental computation strategies.
Key Vocabulary
difference
estimate
Essential Questions
How can we find differences using Base-10 models?
When is an estimate more useful than an exact difference? What are some strategies to
estimate differences?
Why is it important to estimate first, before calculating a difference?
What are different strategies to compute differences? How do we decide which to use?
Essential Understandings
All students should
Understand that estimation skills are valuable and time saving tools particularly in practical situations when exact answers are not required or needed.
Understand that estimation skills are also valuable in determining the reasonableness of
the difference when solving for the exact answer is needed.
Understand that subtraction is used in practical situations when exact answers are
needed.
Develop flexible methods of subtracting whole numbers to find the difference, by
combining numbers in a variety of ways, most depending on place values.
Teacher Notes and Elaborations
Three terms often used in subtraction are:
- 41 subtrahend
57 difference
Subtraction is the inverse operation of addition and is used for different reasons:
– to remove one amount from another;
– to compare one amount to another; and
– to find the missing quantity when the whole quantity and part of the quantity are
known.
The traditional subtraction algorithm for two-digit numbers is contrary to the natural inclination to begin with the left-hand number. One alternative approach to subtraction,
beginning with the left-hand numbers, is finding the partial differences.
98 minuend
- 41 subtrahend
50 begin by subtracting the tens first
+ 7 then subtract the ones
57 then add the partial differences to find the difference
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Computation and Estimation
Virginia SOL 2.7
The student, given two whole numbers,
each of which is 99 or less, will
a. estimate the difference; and b. find the difference, using various
methods of calculation.
Foundational Objectives
1.4
The student, given a familiar problem
situation involving magnitude, will
g. select a reasonable order of
magnitude from three given
quantities: a one-digit numeral, a two-digit numeral, and a three-digit
numeral (e.g., 5, 50, 500); and
h. explain the reasonableness of the
choice.
K.6
The student will model adding and
subtracting whole numbers, using up
to10 concrete items.
Teacher Notes and Elaborations (continued)
Regrouping or trading down is a process of renaming a number to make subtraction easier.
32 minuend 32 equals 20 + 12
- 17 subtrahend 17 equals - (10 + 7)
15 difference 10 + 5 = 15
An understanding of the subtraction algorithm should be built by first using concrete materials and then employing a ―do-and-write‖
approach (i.e., use the manipulatives, then record what you have done). This connects the activity to the written form of the algorithm.
Estimation is a number sense skill used instead of finding an exact answer. When an estimate is needed, the actual computation is not
necessary.
Rounding is one strategy used to estimate.
Estimation is also used before solving a problem to check the reasonableness of the difference when an exact answer is required.
By estimating the result of a subtraction problem, a place value orientation for the answer is established. Front-end estimation may be used
to estimate the difference between two numbers when an exact answer is not required.
Mental computational strategies for subtracting two-digit numbers might include
– lead-digit or front-end strategy: 56 – 21 = ?
50 – 20 = 30
6 – 1 = 5
30 + 5 = 35
– counting up:
87 – 25 = ? or 87 – 25 = ? or 87 – 25 = ?
20 + 60 = 80 25 + 60 = 85 25 + 2 = 27
5 + 2 = 7 85 + 2 = 87 27 + 60 = 87
60 + 2 = 62 60 + 2 = 62 2 + 60 = 62
– partial differences:
98 – 41 = ? 90 – 40 = 50
8 – 1 = 7
50 + 7 = 57
Strategies for subtracting two-digit numbers may include using a hundred chart, number line, and invented strategies.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand
Computation and Estimation
Virginia SOL 2.8
The student will create and solve one-
and two-step addition and subtraction
problems, using data from simple tables, picture graphs, and bar graphs.
Foundational Objectives
1.6
The student will create and solve one-
step story and picture problems using
basic addition facts with sums to 18 or
less and the corresponding subtraction
facts.
K.6
The student will model adding and
subtracting whole numbers, using up
to10 concrete items.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Identify the appropriate data and
operation needed to solve an addition or
subtraction problem where the data are
presented in a simple table, picture
graph, or bar graph.
Solve addition and subtraction problems requiring a one-or two-step
solution, using data from simple tables,
picture graphs, bar graphs, and
everyday life situations.
Create a one-or two-step addition or
subtraction problem using data from
simple tables, picture graphs, and bar
graphs, whose sum is 99 or less.
Key Vocabulary
bar graph data
picture graph
problem
table
Essential Questions
How can we use information from tables and graphs to create and solve addition and
subtraction problems?
How do we know when solving a problem will require more than one step?
How can addition and subtraction help us interpret information from tables and graphs?
Essential Understandings
All students should
Develop strategies for solving practical problems.
Enhance problem-solving skills by creating their own problems.
Teacher Notes and Elaborations
Problem solving means engaging in a task for which a solution or a method of solution is
not known in advance. Solving problems using data and graphs offers a natural way to make
mathematical connections to practical situations.
The ability to retrieve information from simple charts and picture graphs is a necessary
prerequisite to solving problems.
An example of an approach to solving problems is Polya's four-step plan:
– Understand: Retell the problem. – Plan: Decide what operation(s) should be used.
– Solve: Write and solve a number sentence.
– Look back: Does the answer make sense?
The problem solving process is enhanced when students:
– create their own problems; and
– model word problems using manipulatives or drawings.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Computation and Estimation
Virginia SOL 2.9
The student will recognize and describe
the related facts that represent the
inverse relationship between addition and subtraction.
Foundational Objectives
1.5
The student will recall basic addition
facts with sums to 18 or less and the
corresponding subtraction facts.
K.6
The student will model adding and subtracting whole numbers, using up
to10 concrete items.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Determine the missing number in a
number sentence (e.g., 3 + ___ = 5 or
___+ 2 = 5; 5 – __ = 3 or 5 – 2 = ___).
Write the related facts for a given
addition or subtraction fact (e.g., given
3 + 4 = 7, write 4 + 3 = 7, 7 – 4 = 3, and 7 – 3 = 4).
Key Vocabulary
related facts
relationship
Essential Questions
How can an addition fact help us remember a subtraction fact?
What is a fact family?
How can we use models to demonstrate fact families?
How do fact families help us identify missing numbers in number sentences?
Essential Understandings
All students should
Understand how addition and subtraction relate to one another.
Teacher Notes and Elaborations
Addition and subtraction are inverse operations; that is, one undoes the other.
3 + 4 = 7 7 – 3 = 4
7 – 4 = 3 4 + 3 = 7
For each addition fact, there is a related subtraction fact.
Developing strategies for solving missing addends problems and the missing-part of
subtraction facts builds an understanding of the link between addition and subtraction. To
solve 9 – 5 = ? , think 5 + ? = 9.
Demonstrate joining and separating sets to investigate the relationship between addition and
subtraction.
FOCUS K-3______________________STRAND: MEASUREMENT__________________GRADE LEVEL 2
Measurement is important because it helps to quantify the world around us and is useful in so many aspects of everyday life. Students
in grades K–3 should encounter measurement in many normal situations, from their daily use of the calendar and from science
activities that often require students to measure objects or compare them directly, to situations in stories they are reading and to
descriptions of how quickly they are growing.
Measurement instruction at the primary level focuses on developing the skills and tools needed to measure length, weight/mass,
capacity, time, temperature, area, perimeter, volume, and money. Measurement at this level lends itself especially well to the use of
concrete materials. Children can see the usefulness of measurement if classroom experiences focus on estimating and measuring real
objects. They gain deep understanding of the concepts of measurement when handling the materials, making physical comparisons,
and measuring with tools.
As students develop a sense of the attributes of measurement and the concept of a measurement unit, they also begin to recognize the
differences between using nonstandard and standard units of measure. Learning should give them opportunities to apply both
techniques and nonstandard and standard tools to find measurements and to develop an understanding of the use of simple U.S.
Customary and metric units.
Teaching measurement offers the challenge to involve students actively and physically in learning and is an opportunity to tie together
other aspects of the mathematical curriculum, such as fractions and geometry. It is also one of the major vehicles by which
mathematics can make connections with other content areas, such as science, health, and physical education.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Measurement
Virginia SOL 2.10
The student will
a. count and compare a collection of
pennies, nickels, dimes, and quarters whose total value is $2.00
or less; and
b. correctly use the cent symbol (¢),
dollar symbol ($), and decimal
point (.).
Foundational Objectives
1.7
The student will
a. identify the number of pennies equivalent to a nickel, a dime, and a
quarter; and
b. determine the value of a collection
of pennies, nickels, and dimes
whose total value is 100 cents or
less.
K.7
The student will recognize a penny,
nickel, dime, and quarter and will
determine the value of a collection of pennies and/or nickels whose total
value is 10 cents or less.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Determine the value of a collection of
coins and one-dollar bills whose total
value is $2.00 or less.
Compare the values of two sets of coins
and one-dollar bills (each set having a
total value of $2.00 or less), using the terms is greater than, is less than, or is
equal to.
Simulate everyday opportunities to
count and compare, using a collection
of coins and one-dollar bills whose total
value is $2.00 or less.
Use the cent (¢) and dollar ($) symbols
and decimal point (.) to write a value of
money which is $2.00 or less.
Key Vocabulary
bills
cent (¢)
coins
decimal point (.)
dime
dollar ($)
money
nickel
penny
quarter
value
Essential Questions
What strategies help us count a collection of coins?
How can values of coins and one-dollar bills be compared?
What are different ways to write the value of a set of coins or dollars and coins?
When do we count or compare collections of coins and one-dollar bills in real-life
situations?
Essential Understandings
All students should
Understand how to count and compare a collection of coins and one-dollar bills whose total value is $2.00 or less.
Understand the proper use of the cent symbol (¢), dollar sign ($), and decimal point (.).
Teacher Notes and Elaborations
The money system used in the United States consists of coins and bills based on ones, fives,
and tens, making it easy to count money.
The dollar is the basic unit.
Emphasis is placed on the verbal expression of the symbols for cents and dollars (e.g., $0.35 and 35¢ are both read as ―thirty-five cents‖; $3.00 is read as ―three dollars‖).
Counting a collection of coins requires knowledge of place value.
Money can be counted by grouping coins and bills to determine the value of each group and
then added to determine the total value.
The most common way to add amounts of money is to ―count on‖ the amount to be added.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Measurement
Virginia SOL 2.11
The student will estimate and measure
a. length to the nearest centimeter and
inch; b. weight/mass of objects in
pounds/ounces and
kilograms/grams, using a scale; and
c. liquid volume in cups, pints, quarts,
gallons, and liters.
Foundational Objectives
1.9
The student will use nonstandard units
to measure length, weight/mass, and volume.
K.8
The student will identify the
instruments used to measure length
(ruler), weight (scale)…
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Estimate and measure the length of
various line segments and objects to the
nearest inch and centimeter.
Estimate and then measure the
weight/mass of objects to the nearest
pound/ounce and kilogram/gram, using a scale.
Estimate and measure liquid volume in
cups, pints, quarts, gallons, and liters.
Key Vocabulary
centimeter
cup
gallon
gram
inch kilogram
length
liquid volume
liter
mass
measure
metric
ounce
pint
pound
quart
scale US customary
weight
Essential Questions
What types of attributes can be measured?
Why are units used in measuring?
What units and tools are used to measure the attribute of length? How do the units
compare?
How can we estimate and measure the length of various objects?
What units and tools are used to measure the attribute of weight/mass? How do the
units compare?
How can we estimate and measure the weight/mass of various objects?
What units and tools are used to measure the attribute of liquid volume? How do the units compare?
How can we estimate and measure the liquid volume of various containers?
Essential Understandings
All students should
Understand that centimeters/inches are units used to measure length.
Understand how to estimate and measure to determine a linear measure to the nearest
centimeter and inch.
Understand that pounds/ounces and kilograms/grams are units used to measure
weight/mass.
Understand how to use a scale to determine the weight/mass of an object and use the appropriate unit for measuring weight/mass.
Understand that cups, pints, quarts, gallons, and liters are units used to measure liquid
volume.
Understand how to use measuring devices to determine liquid volume in both metric and
customary units.
Teacher Notes and Elaborations
A clear concept of the size of one unit is necessary before one can measure to the nearest
unit.
The experience of making a ruler can lead to greater understanding of using one.
Proper placement of a ruler when measuring length (i.e., placing the end of the ruler at one
end of the item to be measured) should be demonstrated.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Measurement
Virginia SOL 2.11
The student will estimate and measure
a. length to the nearest centimeter and
inch; b. weight/mass of objects in
pounds/ounces and
kilograms/grams, using a scale; and
c. liquid volume in cups, pints, quarts,
gallons, and liters.
Foundational Objectives
1.9
The student will use nonstandard units
to measure length, weight/mass, and
volume.
K.8
The student will identify the
instruments used to measure length
(ruler), weight (scale)…
Teacher Notes and Elaborations (continued)
Weight and mass are different. Mass is the amount of matter in an object. Weight is determined by the pull of gravity on the mass of an
object. The mass of an object remains the same regardless of its location. The weight of an object changes dependent on the gravitational
pull at its location. In everyday life, most people are actually interested in determining an object’s mass, although they use the term weight
(e.g., ―How much does it weigh?‖ versus ―What is its mass?‖).
A balance is a scale for measuring mass. To determine the mass of an object by using a two-pan balance, first level both sides of the
balance by putting standard units of mass on one side to counterbalance the object on the other; then find the sum of the standard units of
mass required to level the balance.
Benchmarks of common objects need to be established for one pound and one kilogram. Practical experience measuring the mass of
familiar objects helps to establish benchmarks.
Pounds and kilograms are not compared at this level.
Knowledge of the exact relationships within the metric or U.S. Customary system of measurement for measuring liquid volume, such as
4 cups to a quart, is not required at this grade level.
Practical experiences measuring liquid volume, using a variety of actual measuring devices (e.g., containers for a cup, pint, quart, gallon,
and liter), will help students build a foundation for estimating liquid volume with these measures.
Volume or capacity is the amount of space or liquid a three-dimensional object or container can hold.
The terms cups, pints, quarts, gallons, and liters are introduced as terms used to describe the liquid volume of everyday containers.
The exact relationship between a quart and a liter is not expected at this level.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Measurement
Virginia SOL 2.12
The student will tell and write time to
the nearest five minutes, using analog
and digital clocks.
Foundational Objectives
1.8
The student will tell time to the half-
hour, using analog and digital clocks.
K.8
The student will identify the
instruments used to measure…time
(clock: digital and analog)…
K.9
The student will tell time to the hour,
using analog and digital clocks.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Show, tell, and write time to the nearest
five minutes, using an analog and
digital clock.
Match a written time to a time shown
on a clock face to the nearest five
minutes.
Key Vocabulary
analog clock
clock
digital clock
hours
minute
time
Essential Questions
What units of time are represented on clocks?
How is reading time on an analog clock different from reading time on a digital clock?
How is it similar?
How does counting by fives help us read time on an analog clock?
How are fractions used in naming certain times?
How can we represent specific times on an analog clock face? …on a digital clock
display?
Essential Understandings
All students should
Apply an appropriate technique to determine time to the nearest five minutes, using
analog and digital clocks.
Demonstrate an understanding of counting by fives to predict five minute intervals when
telling time to the nearest five minutes.
Teacher Notes and Elaborations
Telling time requires reading a clock. The position of the two hands on an analog clock is
read to tell the time. A digital clock shows the time by displaying the time in numbers
which are read as the hour and minutes.
The use of a demonstration clock with gears insures that the position of the hour hand and
the minute hand is precise at all times.
The face of an analog clock can be divided into four equal parts, called quarter hours, of 15
minutes each.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Measurement
Virginia SOL 2.13
The student will
a. determine past and future days of
the week; and b. identify specific days and dates on a
given calendar.
Foundational Objectives
1.11 The student will use calendar language
appropriately (e.g., names of the
months, today, yesterday, next week,
last week).
K.8
The student will identify the
instruments used to measure…calendar
(day, month, and season)…
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Determine the days/dates before and
after a given day/date.
Determine the day that is a specific
number of days or weeks in the past or
in the future from a given date, using a
calendar.
Identify specific days and dates (e.g.,
the third Monday in a given month or
on what day of the week May 11 falls).
Key Vocabulary
after
before
date
day
future last week
month
next week
past
today
week
yesterday
Essential Questions
What units of time are represented on calendars?
How is a calendar organized? How can we find specific dates?
How can we use the calendar to find and describe past and future dates?
Essential Understandings
All students should
Understand how to use a calendar as a way to measure time.
Teacher Notes and Elaborations The calendar is a way to represent units of time (e.g., days, weeks, months).
Using a calendar develops the concept of day as a 24-hour period rather than a period of
time from sunrise to sunset.
Practical situations are appropriate to develop a sense of the interval of time between events
(e.g., Boy Scout meetings occur every week on Monday: there is a week between meetings).
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Measurement
Virginia SOL 2.14
The student will read the temperature
on a Celsius and/or Fahrenheit
thermometer to the nearest 10 degrees.
Foundational Objective
K.8 The student will identify the
instruments used to…temperature
(thermometer).
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Read temperature to the nearest 10
degrees from real Celsius and
Fahrenheit thermometers and from
physical models (including pictorial
representations) of such thermometers.
Key Vocabulary
Celsius (C)
degrees Celsius (°C)
degrees Fahrenheit (°F)
Fahrenheit (F)
temperature
thermometer
Essential Questions
What units and tools are used to measure temperature?
How do we read temperature from a Celsius or Fahrenheit thermometer?
Essential Understandings
All students should
Understand how to measure temperature in Celsius and Fahrenheit with a thermometer.
Teacher Notes and Elaborations
A thermometer is a tool used to measure temperature. A scale marked in every ten degrees Fahrenheit or Celsius indicates temperature. Measurements given in Celsius and Fahrenheit
are two different representations for the same temperature.
The symbols for degrees in Celsius (°C) and degrees in Fahrenheit (°F) should be used to
write temperatures.
Celsius and Fahrenheit temperatures should be related to everyday occurrences by
measuring the temperature of the classroom, the outside, liquids, body temperature, and
other things found in the environment.
Estimating and measuring temperatures in the environment in Fahrenheit and Celsius
requires the use of real thermometers.
A physical model can be used to represent the temperature determined by a real
thermometer.
FOCUS K-3_____________________STRAND: GEOMETRY______________________GRADE LEVEL 2
Children begin to develop geometric and spatial knowledge before beginning school, stimulated by the exploration of figures and
structures in their environment. Geometric ideas help children systematically represent and describe their world as they learn to
represent plane and solid figures through drawing, block constructions, dramatization, and verbal language.
The focus of instruction at this level is on
observing, identifying, describing, comparing, contrasting, and investigating solid objects and their faces;
sorting objects and ordering them directly by comparing them one to the other;
describing, comparing, contrasting, sorting, and classifying figures; and
exploring symmetry, congruence, and transformation.
In the primary grades, children begin to develop basic vocabulary related to these figures but do not develop precise meanings for
many of the terms they use until they are thinking beyond Level 2 of the van Hiele theory (see below).
The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring
student experiences that should lead to conceptual growth and understanding.
Level 0: Pre-cognition. Geometric figures are not recognized. For example, the students cannot differentiate between three-sided
and four-sided polygons.
Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships
between components of a figure. Students should recognize and name figures and distinguish a given figure from others that look
somewhat the same. (This is the expected level of student performance during grades K and 1.)
Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of
geometric figures. (Students are expected to transition to this level during grades 2 and 3.)
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Geometry
Virginia SOL 2.15
The student will
a. draw a line of symmetry in a figure;
and
b. identify and create figures with at
least one line of symmetry.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections and
representations to
Identify figures with at least one line of symmetry, using various concrete
materials.
Draw a line of symmetry – horizontal,
vertical and diagonal – in a figure.
Create figures with at least one line of
symmetry using various concrete
materials.
Key Vocabulary
figure line
symmetry
Essential Questions
What is a line of symmetry? Do all figures have one? Can some figures have more than
one line of symmetry?
What strategies can we use to determine whether a figure has a line of symmetry?
How can we create a figure with a line of symmetry?
Essential Understandings
All students should
Develop strategies to determine whether or not a figure has at least one line of symmetry.
Develop strategies to create figures with at least one line of symmetry.
Understand that some figures may have more than one line of symmetry.
Teacher Notes and Elaborations
A figure is symmetric along a line when one half of a figure is the mirror image of the
other half.
A line of symmetry divides a symmetrical figure, object, or an arrangement of objects into
two parts that are congruent if one part is reflected over the line of symmetry.
Children learn about symmetry through hands-on experiences with geometric figures and the
creation of geometric pictures and patterns.
Guided explorations of the study of symmetry using mirrors, Miras, paper folding, and
pattern blocks will enhance student’s understanding of the attributes of symmetrical figures.
While investigating symmetry, children move figures, such as pattern blocks, intuitively,
thereby exploring transformations of those figures. A transformation is the movement of a
figure – either a translation, rotation, or reflection. A translation is the result of sliding a
figure in any direction; rotation is the result of turning a figure around a point or a vertex;
and reflection is the result of flipping a figure over a line.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Geometry
Virginia SOL 2.16
The student will identify, describe,
compare, and contrast plane and solid
geometric figures (circle/sphere, square/cube, and rectangle/rectangular
prism).
Foundational Objectives
1.12
The student will identify and trace,
describe, and sort plane geometric
figures (triangle, square, rectangle, and
circle) according to number of sides,
vertices, and right angles.
K.11
The student will
a. identify, describe, and trace plane
geometric figures (circle, triangle,
square, and rectangle); and
b. compare the size (larger, smaller)
and shape of plane geometric
figures (circle, triangle, square, and
rectangle).
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Determine similarities and differences
between related plane and solid figures
(e.g., circle/sphere, square/cube, and
rectangle/rectangular prism), using
models and cutouts.
Trace faces of solid figures (e.g., cube and rectangular solid) to create the set
of plane figures related to the solid
figure.
Identify and describe plane and solid
figures (e.g., circle/sphere, square/cube,
and rectangle/rectangular prism),
according to the number and shape of
their faces, edges, and vertices using
models.
Compare and contrast plane and solid
geometric figures (e.g., circle/sphere, square/cube; and rectangle/rectangular
prism) according to the number and
shape of their faces (sides, bases),
edges, vertices, and angles.
Key Vocabulary
angle base
circle cube
edge face
plane rectangle
rectangular prism side solid sphere
square vertex
Essential Questions
How are plane geometric figures different from solid geometric figures? Can they be
similar in any ways?
What are the attributes that determine or identify a solid geometric figure? (faces—sides
and bases, edges, vertices, and angles)
How does a circle compare to a sphere? …a square to a cube? …a rectangle to a
rectangular prism?
How can we find related plane figure(s) by tracing models of solids?
Essential Understandings
All students should
Understand the differences between plane and solid figures while recognizing the
interrelatedness of the two.
Understand that a solid figure is made up of a set of plane figures.
Teacher Notes and Elaborations
An important part of geometry is naming and describing figures in two-dimensions (plane
figures) and three-dimensions (solid figures).
A vertex is a point where two or more line segments, lines, or rays meet to form an angle.
An angle is two rays that share an endpoint.
Plane figures are two-dimensional figures formed by lines that are curved, straight, or a
combination of both. They have angles and sides.
The identification of plane and solid figures is accomplished by working with and handling
objects.
Tracing faces of solid figures is valuable to understanding the set of plane figures related to
the solid figure (e.g., a cube and its square faces or a rectangular prism and its rectangular
faces).
A circle is a closed curve in a plane with all its points the same distance from the center.
A sphere is a (solid) figure with all of its points the same distance from the center.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Geometry
Virginia SOL 2.16
The student will identify, describe,
compare, and contrast plane and solid
geometric figures (circle/sphere, square/cube, and rectangle/rectangular
prism).
Foundational Objectives
1.12
The student will identify and trace,
describe, and sort plane geometric
figures (triangle, square, rectangle, and
circle) according to number of sides,
vertices, and right angles.
K.11
The student will
c. identify, describe, and trace plane
geometric figures (circle, triangle,
square, and rectangle); and
d. compare the size (larger, smaller)
and shape of plane geometric
figures (circle, triangle, square, and
rectangle).
Teacher Notes and Elaborations (continued)
A square is a rectangle with four sides of equal length.
A rectangular prism is a solid in which all six faces are rectangles. A rectangular prism has 8 vertices and 12 edges.
A cube is a (solid) figure with six congruent square faces. All edges are the same length. A cube has 8 vertices and 12 edges. It is a special
type of rectangular prism.
A rectangle is a plane figure with four right angles. A square is a special rectangle.
An edge is the line segment where two faces of a solid figure intersect.
A face is a polygon that serves as one side of a solid figure (e.g., a square is a face of a cube).
A base is a special face of a (solid) figure.
The relationship between plane and solid geometric figures such as the square and the cube or the rectangle and the rectangular prism
helps build the foundation for future geometric study of faces, edges, angles, and vertices.
FOCUS K-3________________STRAND: PROBABILITY AND STATISTICS_________GRADE LEVEL 2
Students in the primary grades have a natural curiosity about their world, which leads to questions about how things fit together or
connect. They display their natural need to organize things by sorting and counting objects in a collection according to similarities and
differences with respect to given criteria.
The focus of probability instruction at this level is to help students begin to develop an understanding of the concept of chance. They
experiment with spinners, two-colored counters, dice, tiles, coins, and other manipulatives to explore the possible outcomes of
situations and predict results. They begin to describe the likelihood of events, using the terms impossible, unlikely, equally likely, more
likely, and certain.
The focus of statistics instruction at this level is to help students develop methods of collecting, organizing, describing, displaying, and
interpreting data to answer questions they have posed about themselves and their world.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Probability and Statistics
Virginia SOL 2.17
The student will use data from
experiments to construct picture graphs,
pictographs, and bar graphs.
Foundational Objectives
1.14
The student will investigate, identify,
and describe various forms of data
collection (e.g., recording daily
temperature, lunch count, attendance,
and favorite ice cream), using tables,
picture graphs, and object graphs.
K.13
The student will gather data by
counting and tallying.
K.14
The student will display gathered data
in object graphs, picture graphs, and
tables, and will answer questions
related to the data.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Read the information presented
horizontally and vertically on picture
graphs, pictographs, and bar graphs.
Collect no more than 16 pieces of data
to answer a given question.
Organize data from experiments, using lists, tables, objects, pictures, tally
marks, symbols and charts, in order to
construct a graph.
Represent data from experiments by
constructing picture graphs,
pictographs, and bar graphs.
Label the axes on a bar graph, limiting
the number of categories (categorical
data) to four and the increments to
multiples of whole numbers
(e.g., multiples of 1, 2, or 5).
On a pictograph, limit the number of
categories to four and include a key
where appropriate.
Key Vocabulary
bar graphs
category
data
experiment
pictograph
picture graph symbol
Essential Questions
Why are picture graphs, pictographs, and bar graphs useful?
What are the characteristics of picture graphs? …pictographs? …bar graphs?
Why do we organize data from experiments into lists, tables, tallies, pictures, symbols,
and/or charts before we create a graph?
How do we construct a picture graph? …pictograph? …bar graph?
Essential Understandings
All students should
Understand that data may be generated from experiments.
Understand how data can be collected and organized in picture graphs, pictographs, and
bar graphs.
Understand that picture graphs use pictures to show and compare data.
Understand that pictographs use a symbol of an object, person, etc.
Understand that bar graphs can be used to compare categorical data.
Teacher Notes and Elaborations
The purpose of a graph is to represent data gathered to answer a question.
Picture graphs are graphs that use pictures to show and compare information. An example
of a picture graph is:
Our Favorite Pets
Cat Dog Horse Fish
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Probability and Statistics
Virginia SOL 2.17
The student will use data from
experiments to construct picture graphs,
pictographs, and bar graphs.
Foundational Objectives
1.14
The student will investigate, identify,
and describe various forms of data
collection (e.g., recording daily
temperature, lunch count, attendance,
and favorite ice cream), using tables,
picture graphs, and object graphs.
K.13
The student will gather data by
counting and tallying.
K.14
The student will display gathered data
in object graphs, picture graphs, and
tables, and will answer questions
related to the data.
Teacher Notes and Elaborations (continued)
Pictographs are graphs that use symbols to show and compare information. A student can be represented as a stick figure in a pictograph.
A key should be used to indicate what the symbol represents (e.g., one picture of a sneaker represents five sneakers in a graph of shoe
types). An example of a pictograph is:
Our Favorite Pets
Cat Dog Horse Fish
= 1 student
Bar graphs are used to compare counts of different categories (categorical data). Using grid paper may ensure more accurate graphs.
– A bar graph uses parallel, horizontal or vertical bars to represent counts for several categories. One bar is used for each category,
with the length of the bar representing the count for that category.
– There is space before, between, and after the bars. The width of the bars should be approximately equal to this space. – The axis displaying the scale that represents the count for the categories should extend one increment above the greatest recorded
piece of data. Second grade students should be collecting data that are recorded in increments of whole numbers, usually
multiples of 1, 2, or 5.
– Each axis should be labeled, and the graph should be given a title.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Probability and Statistics
Virginia SOL 2.18
The student will use data from
experiments to predict outcomes when
the experiment is repeated.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Conduct probability experiments, using
multi-colored spinners, colored tiles, or
number cubes and use the data from the
experiments to predict outcomes if the
experiment is repeated.
Record the results of probability experiments using tables, charts, and
tally marks.
Interpret the results of probability
experiments (e.g., the two-colored
spinner landed on red 5 out of 10
times).
Predict which of two events is more
likely to occur if an experiment is
repeated.
Key Vocabulary
as likely as
certain
chart
equally likely
impossible
likely
outcome
predict
table
tally mark
unlikely
Essential Questions
How do we record and interpret data from a probability experiment?
Why is it important to collect data from many tries?
How can we predict what is likely to happen if the experiment is repeated?
Essential Understandings
All students should
Understand that data may be generated from experiments.
Understand that the likelihood of an event occurring is to predict the probability of it
happening.
Teacher Notes and Elaborations
A spirit of investigation and exploration should permeate probability instruction where
students are actively engaged in investigations and have opportunities to use manipulatives.
Investigation of experimental probability is continued through informal activities, such as
dropping a two-colored counter (usually a chip that has a different color on each side), using
a multicolored spinner (a circular spinner that is divided equally into two, three, four or
more equal ―pie‖ parts where each part is filled with a different color), using spinners with
numbers, or rolling random number generators (dice).
Probability is the chance of an event occurring (e.g., the landing on a particular color when
flipping a two-colored chip is21 , representing one of two possible outcomes).
An event is a possible outcome in probability. Simple events include the possible outcomes
when tossing a coin (heads or tails), when rolling a random number generator (a number
cube or a die where there are six equally likely outcomes and the probability of one
outcome is61 ), or when spinning a spinner.
If all the outcomes of an event are equally likely to occur, the probability of an event is
equal to the number of favorable outcomes divided by the total number of possible
outcomes:
number of favorable outcomes
total number of possible outcomes
At this level, students need to understand only this fractional representation of probability
(e.g., the probability of getting heads when flipping a coin is 21 ).
the probability of the event =
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Probability and Statistics
Virginia SOL 2.18
The student will use data from
experiments to predict outcomes when
the experiment is repeated.
Teacher Notes and Elaborations (continued)
Students should have opportunities to describe in informal terms (i.e., impossible, unlikely, as likely as, equally likely, likely, and certain)
the degree of likelihood of an event occurring. Activities should include practical examples.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Probability and Statistics
Virginia SOL 2.19
The student will analyze data displayed
in picture graphs, pictographs, and bar
graphs.
Foundational Objectives
1.14 The student will investigate, identify,
and describe various forms of data
collection (e.g., recording daily
temperature, lunch count, attendance,
favorite ice cream), using tables, picture
graphs, and object graphs.
1.15
The student will interpret information
displayed in a picture or object graph,
using the vocabulary more, less, fewer,
greater than, less than, and equal to.
K.14
The student will display gathered data
in object graphs, picture graphs, and
tables, and will answer questions
related to the data.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Analyze information from simple
picture graphs, pictographs, and bar
graphs by writing at least one statement
that covers one or both of the
following:
– Describe the categories of data and the data as a whole (e.g., the total
number of responses).
– Identify parts of the data that have
special characteristics, including
categories with the greatest, the
least, or the same.
Select the best analysis of a graph from
a set of possible analyses of the graph.
Key Vocabulary
analyze
category greatest
least
same
Essential Questions
What special features of a graph help us read and interpret it?
What can we tell about the data by studying the graph?
How do we read and interpret a picture graph? …pictograph? …bar graph?
How can we analyze the information in a graph to answer questions, draw conclusions,
and make predictions?
Essential Understandings
All students should
Understand how to read the key used in a graph to assist in the analysis of the displayed data.
Understand how to interpret data in order to analyze it.
Understand how to analyze data in order to answer the questions posed, make
predictions, and generalizations.
Teacher Notes and Elaborations
Statements that represent an analysis and interpretation of the characteristics of the data in
the graph (e.g., similarities and differences, least and greatest, the categories, and total
number of responses) should be discussed with students and written.
When data are displayed in an organized manner, the results of investigations can be described, and the questions posed can be answered.
FOCUS K-3_________STRAND: PATTERNS, FUNCTIONS, AND ALGEBRA________GRADE LEVEL 2
Stimulated by the exploration of their environment, children begin to develop concepts related to patterns, functions, and algebra
before beginning school. Recognition of patterns and comparisons are important components of children’s mathematical development.
Students in kindergarten through third grade develop the foundation for understanding various types of patterns and functional
relationships through the following experiences:
sorting, comparing, and classifying objects in a collection according to a variety of attributes and properties;
identifying, analyzing, and extending patterns;
creating repetitive patterns and communicating about these patterns in their own language;
analyzing simple patterns and making predictions about them;
recognizing the same pattern in different representations;
describing how both repeating and growing patterns are generated; and
repeating predictable sequences in rhymes and extending simple rhythmic patterns.
The focus of instruction at the primary level is to observe, recognize, create, extend, and describe a variety of patterns. These students
will experience and recognize visual, kinesthetic, and auditory patterns and develop the language to describe them orally and in
writing as a foundation to using symbols. They will use patterns to explore mathematical and geometric relationships and to solve
problems, and their observations and discussions of how things change will eventually lead to the notion of functions and ult imately to
algebra.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Patterns, Functions, and Algebra
Virginia SOL 2.20
The student will identify, create, and
extend a wide variety of patterns.
Foundational Objectives
1.17
The student will recognize, describe,
extend, and create a wide variety of
growing and repeating patterns.
K.16
The student will identify, describe, and
extend repeating patterns.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Identify a growing and/or repeating
pattern from a given sequence of
numbers or geometric figures.
Predict the next number, geometric
figure, symbol, picture, or object in a
given pattern. Extend a given pattern, using numbers,
geometric figures, symbols, pictures, or objects.
Create a new pattern, using numbers,
geometric figures, pictures, symbols,
or objects.
Recognize the same pattern in different
manifestations.
Key Vocabulary
difference
growing patterns
object
pattern
predict
repeating pattern
symbol
Essential Questions
What is a pattern?
How can we recognize and identify repeating patterns formed by numbers, geometric
figures, symbols, pictures, or objects?
How can we recognize and identify growing patterns formed by numbers, geometric
figures, symbols, pictures, or objects?
How is a repeating pattern created? …a growing pattern?
How can we analyze a pattern to predict what comes next? How can we extend a
pattern?
How can we recognize a pattern, even when it appears in different forms?
Essential Understandings
All students should
Understand patterns are a way to recognize order and to predict what comes next in an
arrangement.
Analyze how both repeating and growing patterns are generated.
Teacher Notes and Elaborations
Identifying and extending patterns is an important process in mathematical thinking.
Analysis of patterns in the real world (e.g., patterns on a butterfly’s wings, patterns on a
ladybug’s shell) leads to the analysis of mathematical patterns such as number patterns and patterns involving geometric figures.
Reproduction of a given pattern in a different manifestation, using symbols and objects,
lays the foundation for writing numbers symbolically or algebraically.
The simplest types of patterns are repeating patterns. Opportunities to create, recognize,
describe, and extend repeating patterns are essential to the primary school experience.
Growing patterns are more difficult for students to understand than repeating patterns
because not only must they determine what comes next, they must also begin the process
of generalization. Students need experiences with growing patterns in both numerical and
geometry formats.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Patterns, Functions, and Algebra
Virginia SOL 2.20
The student will identify, create, and
extend a wide variety of patterns.
Foundational Objectives
1.17
The student will recognize, describe,
extend, and create a wide variety of
growing and repeating patterns.
K.16
The student will identify, describe, and
extend repeating patterns.
Teacher Notes and Elaborations (continued)
In numeric patterns, students must determine the difference, called the common difference, between each succeeding number in order to
determine what is added to each previous number to obtain the next number. Students should create arithmetic number patterns. Sample
numeric patterns include:
6, 9, 12, 15, 18, (growing pattern grows by 3);
20, 18, 16, 14, (growing pattern grows by -2);
1, 2, 4, 7, 11, 16, (growing pattern grows by 1, then by 2, then by 3, etc); and 1, 3, 5, 1, 3, 5, 1, 3, 5… (repeating pattern with a core of 1, 3, 5).
In geometry patterns, students must often recognize transformations of a figure, particularly rotation or reflection. Rotation is the result of
turning a figure around a point or a vertex, and reflection is the result of flipping a figure over a line.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Patterns, Functions, and Algebra
Virginia SOL 2.21
The student will solve problems by
completing numerical sentences
involving the basic facts for addition and subtraction. The student will create
story problems, using the numerical
sentences.
Foundational Objective
1.18
The student will demonstrate an
understanding of equality through the
use of the equal sign.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Solve problems by completing a
numerical sentence involving the basic
facts for addition and subtraction,
e.g., 3 + __ = 7, or 9 – __ = 2.
Create a story problem for a given
numerical sentence.
Key Vocabulary
number sentence
story problem
Essential Questions
How can we use models to represent an addition or subtraction situation?
How can we create a story problem from a basic fact or numerical sentence?
How can we write number sentences to represent problems with missing quantities?
What basic facts do these number sentences represent?
How can the relationship between addition and subtraction be used to complete number
sentences and solve problems?
Essential Understandings
All students should
Use mathematical models to represent and understand quantitative relationships.
Understand various meanings of addition and subtraction and the relationship between
the two operations.
Understand how to write missing addend and missing subtrahend sentences.
Teacher Notes and Elaborations
Recognizing and using patterns and learning to represent situations mathematically are
important aspects of primary mathematics.
Discussing what a word problem is saying and writing a number sentence are precursors to
solving word problems.
The patterns formed by related basic facts facilitate the solution of problems involving a
missing addend in an addition sentence or a missing part (subtrahend) in a subtraction
sentence.
Making mathematical models to represent simple addition and subtraction problems
facilitates their solution.
By using story problems and numerical sentences, students begin to explore forming
equations and representing quantities using variables.
Students can begin to understand the use of a symbol (e.g., __, ?, or ) to represent an unknown quantity.
GRADE 2 CURRICULUM GUIDE PRINCE WILLIAM COUNTY SCHOOLS
Curriculum Information
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaborations
Strand Patterns, Functions, and Algebra
Virginia SOL 2.22
The student will demonstrate an
understanding of equality by
recognizing that the symbol ―=‖ in an equation indicates equivalent
quantities and the symbol ―≠‖ indicates
that quantities are not equivalent.
Foundational Objective
1.18
The student will demonstrate an
understanding of equality through the
use of the equal sign.
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections
and representations to
Identify the equality (=) and inequality
(≠) symbols.
Identify equivalent values and
equations (e.g., 8 = 8 and 8 = 4 + 4).
Identify nonequivalent values and
equations (e.g., 8 ≠ 9 and 4 + 3 ≠ 8).
Identify and use the appropriate symbol
to distinguish between equal and not
equal quantities (e.g., 8 + 2 = 7 + 3 and
1 + 4 ≠ 6 + 2).
Key Vocabulary
equality symbol (=)
equality
equivalent
inequality symbol (≠)
Essential Questions
What does the equals sign (=) tell about a number sentence (equation)?
When is the symbol ≠ used in number sentences?
How can we test the truth of a number sentence?
Essential Understandings
All students should
Understand that the equal symbol means equivalent (same as) quantities.
Understand that the inequality symbol (≠) means the values on both sides are not
equivalent.
Teacher Notes and Elaborations
The ―=‖ symbol means that the values on either side are the same (balanced).
The ―≠‖ symbol means that the values on either side are not the same (not balanced).
In order for students to develop the concept of equality, students need to see the ―=‖ symbol
used in various locations in an equation (e.g., 3 + 4 = 7 and 5 = 2 + 3).
A number sentence is an equation with numbers (e.g., 6 + 3 = 9; or 6 + 3 = 4 + 5).